Cyclic Vectors in Banach Spaces of Analytic Functions

Part of the NATO ASI series book series (ASIC, volume 153)


In these three lectures we consider Banach spaces of analytic functions on plane domains. If the space admits the operator of multiplication by z, then it is of interest to describe the cyclic vectors for this operator, that is, those functions in the space with the property that the polynomial multiples of the function are dense. A necessary condition is that the function have no zeros; in general it is difficult to give necessary and sufficient conditions.


Banach Space Analytic Function Invariant Subspace Bergman Space Blaschke Product 
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  1. [1]
    Aharonov, A., Shapiro, H.S. and Shields, A.L., “Weakly inver-tible elements in the space of square-summable holomorphic functions”, J. London Math. Soc. (2) 9 (1974/5), pp. 183–192.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Beauzamy, B., “Unoperateur sans sous-espace invariant; simplification de l’example de P. Enflo”, preprint.Google Scholar
  3. [3]
    Bercovici, H., Foias, C., and Pearcy, C.M., “Invariant sub-spaces, dilation theory, and dual algebras”, CBMS regional conference series in mathematics, Amer. Math. Soc, Providence, R.I., to appear.Google Scholar
  4. [4]
    Berman, R., Brown, L. and Conn, W., “Cyclic vectors of bounded characteristics in Bergman spaces”, preprint.Google Scholar
  5. [5]
    Beurling, A., “Ensembles exceptionnels”, Acta Math. 72 (1940), pp. 1–13.MathSciNetCrossRefGoogle Scholar
  6. [6]
    Beurling, A., “On two problems concerning linear transformations in Hilbert space”, Acta Math. 81 (1949), pp. 239–255.zbMATHCrossRefGoogle Scholar
  7. [7]
    Bourdon, P., “Cyclic Nevanlinna class functions in Bergman spaces”, preprint.Google Scholar
  8. [8]
    Brown, L. and Cohn, W., “Some examples of cyclic vectors in the Dirichelt space”, preprint.Google Scholar
  9. [9]
    Brown, L. and Shields, A.L., “Cyclic vectors in the Dirichlet space”, Trans. Amer. Math. Soc. 285 (1984), pp. 269–304.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Carleson, L., “On a class of meromorphic functions and its associated exceptional sets”, Appelbergs Boktryckeri, Uppsala, 1950.zbMATHGoogle Scholar
  11. [11]
    Carleson, L., “Sets of uniqueness for functions regular in the unit circle”, Acta Math. 87 (1952), pp. 325–345.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Carleson, L., “A representation formula for the Dirichlet integral”, Math. Zeit. 73 (1960), pp. 190–196.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    Caughran, J., “Two results concerning the zeros of functions with finite Dirichlet integral”, Can. J. Math. 21 (1969), pp. 312–316.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    Duren, P., Theory of H p spaces, Pure and Appl. Math., vol. 38, Academic Press, New York (1970).zbMATHGoogle Scholar
  15. [15]
    Duren, P., Romberg, B.W. and Shields, A.L., “Linear functionals on Hp spaces with 0 < p < 1”, J. reine angew. Math. 238 (1969), pp. 32–60.MathSciNetzbMATHGoogle Scholar
  16. [16]
    Enflo, P., “On the invariant subspace problem in Banach spaces”, Acta. Math., to appear.Google Scholar
  17. [17]
    Garnett, J., Bounded analytic functions, Pure and Appl. Math., vol. 96, Academic Press, New York (1981).zbMATHGoogle Scholar
  18. [18]
    Hartman, P., and Kershner, R., “The structure of monotone functions”, Amer. J. Math. 59 (1937), pp. 809–822.MathSciNetCrossRefGoogle Scholar
  19. [19]
    Hayman, W., “On Nevanlinna’s second theorem and extensions”, Rend. Circ. Mat. Palrmo (2)2 (1953), pp. 346–392.MathSciNetCrossRefGoogle Scholar
  20. [20]
    Hoffman, K., Banach spaces of analytic functions, Prentice Hall, Englewood Cliffs, New Jersey (1962).zbMATHGoogle Scholar
  21. [21]
    Horowitz, C., “Zeros of functions in Bergman spaces”, Duke Math. J. 41 (1974), pp. 693–710.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    Kahane, J., Some random series of functions, D.C. Heath and Co., Lexington, Mass. 1968.zbMATHGoogle Scholar
  23. [23]
    Kelley, J., General Topology, van Nostrand Co., New York, 1955.zbMATHGoogle Scholar
  24. [24]
    Kinney, J., “Tangential limits of functions of the class Sα”, Proc. Amer. Math. Soc. 14 (1963), pp. 68–70.MathSciNetzbMATHGoogle Scholar
  25. [25]
    Koosis, P., Introduction to H p spaces, London Math. Soc, Lecture Note Series 40, Cambridge Univ. Press, London, 1980.zbMATHGoogle Scholar
  26. [26]
    Kopp, R.P., “A subcollection of algebras in a collection of Banach spaces”, Pac. J. Math. 30 (1969), pp. 433–435.MathSciNetzbMATHGoogle Scholar
  27. [27]
    Korenblum, B.I., “Functions holomorphic in a disc and smooth in its closure”, Doklady Akad. Nauk. SSSR 200 (1971), pp. 24–27; English translation: Soviet Math. Doklady 12 (1971), pp. 1312–1315.Google Scholar
  28. [28]
    Korenblum, B.I., “Invariant subspaces of the shift operator in a weighted Hilbert space”, Matem. Sbornik 89 (131) (1972), pp. 110–137; English translation: Math. USSR Sbornik 18 (1972), pp. 111–138.MathSciNetGoogle Scholar
  29. [29]
    Korenblum, B.I., “An extension of the Nevanlinna theory”, Acta Math. 135 (1975), pp. 187–219.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    Korenblum, B.I., “A Beurling-type theorem”, Acta. Math. 138 (1977), pp. 265–293.MathSciNetzbMATHCrossRefGoogle Scholar
  31. [31]
    Korenblum, B.I., “Cyclic elements in some spaces of analytic functions”, Bull. Amer. Math. Soc. 5 (1981), pp. 317–318.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [32]
    Landau, E., Darstellung und Begründung einiger neuerer Ergebnisse der Funktionetheorie, Zweite Auflage, J. Springer Verlag, Berlin, 1929.Google Scholar
  33. [33]
    Nagel, A., Rudin, W. and Shapiro, J.H., “Tangential boundary behaviour of harmonic extenstions of Lp potentials”, Conference on Harmonic Analysis in honor of Antoni Zygmund, vol. II, 1982, pp. 533–548, ed. W. Beckner et al, Wadsworth Publishers, Belmont, Calif.Google Scholar
  34. [34]
    Nagel, A., Rudin, W. and Shapiro, J.H., “Tangential boundary behaviour of functions in Dirichlet-type spaces”, Annals of Math. 116 (1982), pp. 331–360.MathSciNetzbMATHCrossRefGoogle Scholar
  35. [35]
    Nelson, J.D., “A characterization of zero sets for A”, Michigan Math. J. 18 (1971), pp. 141–147.MathSciNetzbMATHCrossRefGoogle Scholar
  36. [36]
    Read, C.J., “A solution to the invariant subspace problem”, Bull. London Math. Soc. 16 (1984), 337–401.MathSciNetzbMATHCrossRefGoogle Scholar
  37. [37]
    Read, C.J., “A solution to the invariant subspace problem on the space ℓ1, preprint.Google Scholar
  38. [38]
    Roberts, J., “Cyclic inner functions in the Bergman spaces and weak outer functions in Hp, 0 < p < 1”, Illinois J. Math. (to appear).Google Scholar
  39. [39]
    Salem, R. and Zygmund, A., “Capacity of sets and Fourier series”, Trans. Amer. Math. Soc. 59 (1946), pp. 23–41.MathSciNetzbMATHCrossRefGoogle Scholar
  40. [40]
    Shamoyan, F.A., “Weak invertibility in some spaces of analytic functions”, Akad. Nauk Armyan. SSR Doklady 74 (1982), no. 4, pp. 157–161; MR 84e: 30077.MathSciNetzbMATHGoogle Scholar
  41. [41]
    Shapiro, H.S., “Weakly invertible elements in certain function spaces, and generators in ℓ1”, Mich. Math. J. 11 (1964), pp. 161–165.zbMATHCrossRefGoogle Scholar
  42. [42]
    Shapiro, H.S., “Weighted polynomial approximation and boundary behaviour of analytic functions”, Contemporary Problems in Analytic Functions (Internat. Conference, Erevan, Armenia 1965). “Nauka” Moscow 1966, pp. 326–335.Google Scholar
  43. [43]
    Shapiro, H.S., “Some remarks on weighted polynomial approximation of holomorphic functions”, Mat. Sbornik 73 (115) (1967), pp. 320–330; English translation: Math. USSR Sb. 2 (1967), pp. 285–294.Google Scholar
  44. [44]
    Shapiro, H.S., “Monotone singular functions of high smoothness”, Mich. Math. J. 15 (1968), pp. 265–275.zbMATHCrossRefGoogle Scholar
  45. [45]
    Shapiro, H.S. and Shields, A.L., “On the zeros of functions with finite Dirichlet integral and some related function spaces”, Math. Zeit. 80 (1962), pp. 217–229.MathSciNetzbMATHCrossRefGoogle Scholar
  46. [46]
    Shapiro, J.H., “Cyclic inner functions in Bergman spaces”, preprint (1980), not for publication.Google Scholar
  47. [47]
    Shields, A.L., “Weighted shift operators and analytic function theory”, Math. Surveys 13; Topics in operator theory, ed. C.M. Pearcy, Amer. Math. Soc., Providence, R.I. (1974), pp. 49–128 (second printing, with addendum, 1979).Google Scholar
  48. [48]
    Shields, A.L., “Cyclic vectors in some spaces of analytic functions”, Proc. Royal Irish Acad., 74 Sect. A (1974), pp. 293–296.MathSciNetzbMATHGoogle Scholar
  49. [49]
    Shields, A.L., “Cyclic vectors in spaces of analytic functions”, Issled. po lin. oper. i teor. funkc., Zapiski naucn. Seminar LOMI 8 (1978), pp. 142–144.Google Scholar
  50. [50]
    Shields, A.L., “An analogue of a Hardy-Littlewood-Fejér inequality for upper triangular trace class operators”, Math. Zeitsch. 182 (1983), pp. 473–484.MathSciNetzbMATHCrossRefGoogle Scholar
  51. [51]
    Stegenga, D.A., “Multipliers of the Dirichlet space”, Illinois J. Math. 24 (1980), 113–139.MathSciNetzbMATHGoogle Scholar
  52. [52]
    Taylor, B.A. and Williams, D.L., “Zeros of Lipschitz functions analytic in the unit disc”, Michigan Math. J. 18 (1971), pp. 129–139.MathSciNetzbMATHCrossRefGoogle Scholar
  53. [53]
    Taylor, G.D., “Multipliers on Dα”, Trans. Amer. Math. Soc. 123 (1966), pp. 229–240.MathSciNetzbMATHGoogle Scholar

Copyright information

© D. Reidel Publishing Company 1985

Authors and Affiliations

  1. 1.Department of MathematicsThe University of MichiganAnn ArborUSA

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